Answer
$\dfrac{2x\sqrt[4]{2xy^3}}{y^2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt[4]{\dfrac{32x^5}{y^5}}
,$ make the denominator a perfect power of the index so that the final result will already be in rationalized form. Then find a factor of the radicand that is a perfect power of the index. Finally, extract the root of that factor.
$\bf{\text{Solution Details:}}$
Multiplying the radicand that will make the denominator a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt[4]{\dfrac{32x^5}{y^5}\cdot\dfrac{y^3}{y^3}}
\\\\=
\sqrt[4]{\dfrac{32x^5y^3}{y^8}}
.\end{array}
Factoring the radicand into an expression that is a perfect power of the index and then extracting its root result to
\begin{array}{l}\require{cancel}
\sqrt[4]{\dfrac{16x^4}{y^8}\cdot2xy^3}
\\\\=
\sqrt[4]{\left( \dfrac{2x}{y^2}\right)^4\cdot2xy^3}
\\\\=
\dfrac{2x}{y^2}\sqrt[4]{2xy^3}
\\\\=
\dfrac{2x\sqrt[4]{2xy^3}}{y^2}
.\end{array}
Note that all variables are assumed to have positive values.
